Multidimensional renewal theory in the non-centered case. Application to strongly ergodic Markov chains

0.31 MB·English

by  Denis Guibourg

Multidimensional renewal theory in the non-centered case.
Application to strongly ergodic Markov chains.
2
1
0 D. GUIBOURG and L. HERVÉ ∗
2

n
January 11, 2012
a
J

0
1

AMS subject classification : 60J10-60K05-47A55
]
R
P Keywords : Fourier techniques, spectral method.
.
h
at Abstract. Let(Sn)n≥0beaRd-valuedrandomwalk(d≥2). UsingBabillot’smethod[2],wegivegeneralconditions
m onthecharacteristicfunctionofSnunderwhich(Sn)n 0 satisfiesthesamerenewaltheoremasintheindependentcase
≥
[ (i.e.thesameconclusionasinthecasewhentheincrementsof(Sn)n 0 areassumedtobeindependentandidentically
≥
distributed). This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice
2
v condition and (almost) optimal moment conditions.
3
0
6
1 Introduction
3
.
0
1
1 Let (Sn)n 0 be a Rd-valued random walk. Renewal theory gives the behavior, as a + ,
1 of the pos≥itive measures U () defined on the Borel σ-algebra B(Rd) of Rd as follokwsk→: ∞
a
: ·
v
i +
X a Rd, A B(Rd), U (A) = ∞E 1 (S a) . (1)
a A n
r ∀ ∈ ∀ ∈ −
a nX=1 (cid:2) (cid:3)
TodefinetherenewalmeasureU (),thesequence(S ) hastobetransient: forindependent
a n n 0
· ≥
or Markov random walks, this leads to consider the following cases:
1. d 3 and E[S ]= 0 (centered case),
1
≥
2. d 1 and E[S ] = 0 (non-centered case): in this case, the behavior of U () is specified
1 a
≥ 6 ·
when a + in the direction of E[S ].
1
k k→ ∞
The behavior of U () also depends on the usual lattice or non-lattice conditions.
a
·
This work is the continuation of [11] (case d = 1) and [13] (centered case in dimension
d 3). More specifically, in this paper, we consider the non-centered case in dimension
≥
d 2, and we present some general assumptions involving the characteristic function of S ,
n
≥
∗Université Européenne de Bretagne, I.R.M.A.R. (UMR-CNRS 6625), Institut National des Sciences Ap-
pliquées deRennes. [email protected], [email protected]
1
underwhichwehavethesameconclusionasintheclassicalrenewaltheoremforrandomwalks
with independent and identically distributed (i.i.d.) increments. By using the weak spectral
method [14], this result is then applied to additive functionals of strongly ergodic Markov
chains. This work is greatly inspired by Babillot’s paper [2]. Before presenting our results,
we give a brief review of well-known multidimensional renewal theorems in both independent
and Markov settings, as well as some general comments on Fourier’s method. To that effect
we introduce some notations which will be repeatedly used afterwards:
- , is the canonical scalar product on Rd,
h· ·i
- is the associated euclidean norm on Rd,
k·k
- L () is the Lebesgue-measure on Rd,
d
·
- (Rd,C) is the set of complex-valued continuous compactly supported functions on Rd,
c
C
- the Fourier transform of any Lebesgue-integrable function f : Rd C is defined as follows:
t Rd, fˆ(t) := L (e i t, f), →
d −h ·i
∀ ∈
- is the set of complex-valued continuous Lebesgue-integrable functions on Rd, whose
H
Fourier transform is compactly supported and infinitely differentiable on Rd,
- for any R > 0, we denote by B := B(0,R) the open ball: B(0,R) := t Rd : t < R ,
R
{ ∈ k k }
- for any 0 < r < b, we denote by K the annulus K := t Rd :r < t < b .
r,b r,b
{ ∈ k k }
Renewal theory for random walks with i.i.d. non-centered increments.
Let (X ) be a sequence of i.i.d. non-centered random variables (r.v) taking values in Rd,
n n 1
≥
and let S = X +...+X . In dimension d 2, the renewal theorem was first established
n 1 n
≥
by Ney and Spitzer [19] in the lattice case. Extension to the non-lattice case was obtained by
Doney [6]under Cramer’s condition, andby Stam[23]under theweaker non-lattice condition.
Setting m := max(d 1,2), Stam’s statement writes as follows:
d −2
E X1 md < , m~ := E[X1]= 0 g c(Rd,R), lim τd−21 Uτm~(g) = CLd(g) (2)
k k ∞ 6 ⇒ ∀ ∈C τ +
→ ∞
(cid:2) (cid:3)
where C is a positive constant depending on the first and second moments of X . In the
1
lattice case, Property (2) still holds, but L () must be replaced with the product of counting
d
·
and Lebesgue measures both defined on some sublattices of Rd. Stam’s proof is based on the
1
local limit theorem (LLT) due to Spitzer [22, Th. P7.10] . More precisely, this LLT is applied
to study the difference
+n=∞1n(d−1)/2 E[g(Sn −a)]−E[g(Tn −a)] ,
where the r.v. T are dePfined as the pa(cid:0)rtial sums of a i.i.d. sequenc(cid:1)e of Gaussian r.v. having
n
the same first and second moments as X . Then (2) is deduced from the Gaussian case.
1
Fourier techniques in renewal theory (Breiman’s method).
The weak convergence in (2) can be established by investigating the behavior of U (h) for
a
h . In fact the inverse Fourier formula gives (without any assumption on the model):
∈H
h , E h(S a) = (2π) d hˆ(t)E[ei t,Sn ]e i t,a dt. (3)
∀ ∈ H n− − Rd h i −h i
ThisisthestartingpointofF(cid:2)ourier’s m(cid:3)ethodinprRobability theory. Inthei.i.d.case,using(3)
and the formula E[ei t,Sn ] = E[ei t,X1 ]n, the potential U (h) given by (1) is equal to the
h i h i a
1This LLT, established byFourier techniques,extendsto d≥2 the one-dimensional result of [21].
2
following integral:
U (h) = I(a) := (2π) d hˆ(t) φ(t) e i t,a dt with φ(t) =E[ei t,X1 ], (4)
a − K 1 φ(t) −h i h i
−
R
where K is the support of hˆ. More precisely, since E[X ] = 0 and d 2, the integrand in
1
6 ≥
I(a) is integrable at 0. Thus I(a) is well-defined provided that φ(t) < 1 for all t = 0: this
| | 6
is the non-lattice condition. Fourier’s method also applies to the lattice case by considering a
periodic summation in (4). The renewal theorem then follows from the study of the integrals
I(τm~) when τ + . This method, introduced by Breiman [4] in dimension d = 1, was
→ ∞
extended to d 2 by Babillot [2] in the general setting of Markov random walks (see below).
≥
Renewal theory for Markov random walks.
Let (E, ) denote a measurable space, and let (Xn,Sn)n N be an E Rd-valued Markov
E ∈ ×
random walk (MRW), namely: (Xn,Sn)n N is a Markov chain and its transition kernel P
∈
satisfies the following additive property (in the second component):
(x,s) E Rd, A , B B(Rd), P (x,s),A B = P (x,0),A (B s) . (5)
∀ ∈ × ∀ ∈ E ∀ ∈ × × −
As usual we set S = 0. When (X ) is stro(cid:0)ngly ergodic a(cid:1)nd S(cid:0)is non-centered, B(cid:1)abillot
0 n n 0 1
≥
gives in [2] some (operator-type) moment and non-lattice conditions for the additive compo-
nent(S ) tosatisfytherenewal conclusion in(2). Recallthatthestrongergodicity condition
n n
states that the transition kernel Q of (X ) admits an invariant probability measure π, and
n n 0
≥
that there exists a Banach space ( , ), composed of π-integrable functions on E and
B k · kB
containing the function 1 , such that π defines a continuous linear form on and
E
B
lim sup Qnf π(f)1 = 0. (6)
E
n→+∞f , f 1k − kB
∈B k k≤
The proof in [2] is based on Fourier techniques and the usual Nagaev-Guivarc’h spectral
method involving the semi-group of Fourier operators associated with (Xn,Sn)n N, namely:
∈
n N, t Rd, x E, Q (t)f (x) := E ei t,Sn f(X ) ,
n (x,0) h i n
∀ ∈ ∀ ∈ ∀ ∈
where E denotes the expectation und(cid:0)er the i(cid:1)nitial distribu(cid:2)tion (X ,S )(cid:3) δ . The
(x,0) 0 0 (x,0)
∼
operators Q (t) act (for instance) on the space of bounded measurable functions f : E C.
n
→
The semi-group property writes as follows: (m,n) N2, Q (t) = Q (t) Q (t). In
m+n m n
∀ ∈ ◦
particular we have Q (t) = Q (t)n. This property is the substitute for MRWs of the formula
n 1
E[ei t,Sn ]= E[ei t,X1 ]n of the i.i.d. case.
h i h i
The content of the paper.
Section 2 focuses on Fourier’s method. More specifically we consider a general sequence
(Xn,Sn)n N (not necessarily a MRW) of random variables taking values in E Rd. In sub-
stance our∈non-centered condition writes as follows: m~ := lim E[S ]/n exists×in Rd and is
n n
nonzero. Let f : E [0,+ ) be such that E[f(X )] < for every n 1. Under a general
n
→ ∞ ∞ ≥
hypothesis, called (m), on the functions t E f(X )ei t,Sn , Theorem 1 states that there
n h i
R 7→
exists some positive constant C (specified later) such that we have
(cid:2) (cid:3)
+
g c(Rd,R), τd−21 ∞E f(Xn)g(Sn a) CLd(g)
∀ ∈C − −→
n=1
X (cid:2) (cid:3)
3
when a := a(τ) Rd goes to infinity "around the direction m~" in the sense (specified later)
∈
defined in [24] (for instance, take a := τm~ with τ + ). Actually Hypothesis (m), intro-
→ ∞ R
duced in [13] (centered case), contains the tailor-made conditions to prove renewal theorems
via Fourier’s method. The proof of Theorem 1 borrows the lines of [2] with the following
improvements. First, the distribution-type arguments and the modified Bessel functions used
in [2] are replaced with elementary computations. Second, the (asymmetric) dyadic decom-
position, partially developed in [2, 1] to study integrals of type (4), is detailed in this work.
Section 3 is devoted to the Markov context. Specifically, we assume that (Xn)n N is a
∈
Markov chain satisfying one of the three following classical strong ergodicity assumptions:
- (X ) is ρ-mixing (see [20]),
n n 0
≥
- (X ) is V-geometrically ergodic (see [18]),
n n 0
≥
- (X ) is a strictly contractive Lipschitz iterative model (see [7]).
n n 0
≥
Letξ beaRd-valuedmeasurablefunction,andletS = ξ(X )+...+ξ(X ). Thenthesequence
n 1 n
(Xn,Sn)n N is a special instance of MRW. As already used in [13], the weak spectral method
∈
[14] allows us to reduce Hypothesis (m) to a non-lattice condition and to some (almost)
R
optimal moment conditions on ξ, which are much weaker than those in [2].
Theorem 1 should supply further interesting applications, not only in Markov models but
also in dynamical systems associated with quasi-compact Perron-Frobenius operators. On
that subject, recall that the renewal theorems yield the asymptotic behavior of counting
functions arising in the geometry of groups, as already developed for instance in [17, 5, 24].
2 Renewal theory in the non-centered case (Fourier method)
For any A Rd, g : A C, and τ (0,1], we define the following quantities in [0,+ ]:
⊂ → ∈ ∞
g(x) g(y)
g = sup g(x) and g := sup | − |, (x,y) A2, x = y .
0,A | | τ,A x y τ ∈ 6
x A
∈ k − k
(cid:13) (cid:13) (cid:2) (cid:3) (cid:8) (cid:9)
(cid:13) (cid:13)
We say that g is τ-Hölder on A if [g] < . Moreover, for any open subset of Rd and
τ,A
∞ O
every m N , we denote by m( ,C) the vector space composed of m-times continuously
∈ ∗ Cb O
differentiable functions f : C with bounded partial derivatives on . If m (0,+ ) N,
O→ O ∈ ∞ \
we set τ := m m where m is the integer part of m, and we denote by m( ,C) the
−⌊ ⌋ ⌊ ⌋ Cb O
vector space composed of functions f : C satisfying the three following conditions:
O→
f is m -times continuously differentiable on ,
⌊ ⌋ O
Each partial derivative of order j = 0,..., m of f is bounded on ,
⌊ ⌋ O
Each partial derivative of order m of f is τ-hölder on .
⌊ ⌋ O
Define f := (∂f ) if m 1, and Hessf := ( ∂2f ) (Hessian matrix) if m 2.
∇ ∂xi 1≤i≤d ≥ ∂xi∂xj 1≤i,j≤d ≥
Let (Ω, ,P) be a probability space. We denote by (E, ) a measurable space, and we
F E
considerasequence(X ,S ) ofE Rd-valuedrandomvariablesdefinedonΩ. Throughout,
n n n 0
≥ ×
we assume d 2.
≥
4
Hypothesis (m). Given m [2,+ ) andf :E [0,+ ) a measurable function satisfying
R ∈ ∞ → ∞
n 1, E[f(X )] < , (7)
n
∀ ≥ ∞
we say that Hypothesis (m) holds if the following conditions are fulfilled:
R
(i) There exists R > 0 such that, for all t B and all n 1, we have:
R
∈ ≥
E f(X )ei t,Sn = λ(t)nL(t)+R (t), (8)
n h i n
where λ(0) = 1, the functio(cid:2)ns λ(·) andL((cid:3)·) are inCmb BR,C , andthe series n 1Rn(·)
uniformly converges on the open ball B and defines a function in m B ,C .≥
R (cid:0) (cid:1) Cb R P
(ii) For all 0 < r < b, the series E f(X )ei ,Sn uniformly converge(cid:0)s on th(cid:1)e annulus
n 1 n h· i
K and defines a function in m≥ K ,C .
r,b PCb (cid:2)r,b (cid:3)
(cid:0) (cid:1)
Under Hypothesis (m), we set
R
m~ := i λ(0) and Σ := Hessλ(0).
− ∇ −
Below we assume that m~ = 0: this is our non-centered condition. In fact, under Hypoth-
6
esis (m) and additional mild conditions (see [13, Prop. 1]), we have m~ = lim E[S ]/n,
n n
R
so that m~ may be viewed as a nonzero mean vector in Rd. Below we also assume that the
symmetric matrix Σ is positive-definite. In the Markov setting of Section 3, Σ is linked to
some covariance matrix, see (36).
Hypothesis (H). Setting m := max(d 1,2), there exists a real number m > m such that
d −2 d
Hypothesis (m) holds. We have L(0) = 0, m~ = 0, and Σ := Hessλ(0) is positive-definite.
R 6 6 −
Theorem 1 Assume that Hypothesis (H) holds. Then, for each function a : [0,+ ) Rd
∞ →
such that
a(τ) τm~
A:= lim − exists in Rd, (9)
τ + √τ
→ ∞
the family V (), τ (0,+ ) of positive measures on Rd defined by
τ
{ · ∈ ∞ }
+
A B(Rd), Vτ(A) := (2πτ)d−21 ∞E f(Xn)1A(Sn a(τ)) (10)
∀ ∈ −
n=1
X (cid:2) (cid:3)
weakly converges to CL () as τ + , where C := C(L,m~,Σ,A) (0,+ ) is given by:
d
· → ∞ ∈ ∞
detS Sm~,SA 2 Sm~ 2 SA 2
1
C(L,m~,Σ,A) := L(0) exp −k k k k with S := Σ−2.
Sm~ 2 Sm~ 2
k k (cid:18)(cid:10) (cid:11) k k (cid:19)
Condition (9), introduced in [24], specifies what we called "around the direction m~" in
Introduction. Obviously a(τ) = τm~ satisfies (9).
Remark 1 Theorem 1 may be extended to the lattice case: Hypothesis (m)(ii) must be
R
adapted, and L () is replaced with the product of the counting measure and the Lebesgue
d
·
measure both associated with some sublattices of Rd, see [12, Sect. 2.5].
The next subsections are devoted to the proof of Theorem 1.
5
2.1 Some reductions and Fourier techniques
Change of coordinates. Let ~e denote the first vector of the canonical basis of Rd, and
1
•
let T be any isometric linear map in Rd such that T(m~) = m~ ~e . Up to replace S with
1 n
k k
TS , one may assume without loss of generality that m~ = m~ ~e . This leads to replace
n 1
k k
λ(),L(),R (),h(),Σ with λ T 1,L T 1,R T 1,h T 1,T Σ T 1.
n − − n − − −
· · · · ◦ ◦ ◦ ◦ ◦ ◦
The function w. For all x = (x ,x ,...,x ) Rd, we set x := (x ,...,x ) Rd 1. The
1 2 d ′ 2 d −
• ∈ ∈
following function w() will play an important role:
·
x Rd, w(x) = ix + x 2. (11)
1 ′
∀ ∈ − k k
Remark 2 We have: x Rd, w(x) x 3/4 x 1/2. Thus 1/w is integrable at 0.
1 ′
∀ ∈ | | ≥ | | k k
Remark 3 Somesimplefacts on the function λ()of (8) can be deduced from Hypothesis (H).
·
First, since m 2, we have
d
≥
λ(t) = 1+i m~ t 1 Σt,t +o( t 2). (12)
k k 1− 2h i k k
Second, since L() is continuous on B and L(0) = 0, one may suppose (up to reduce R) that
R
· 6
we have: t B , L(t) = 0. By Hypothesis (m)(ii), the last property then implies that, for
R
∀ ∈ 6 R
all t B 0 , the series λ(t)n converges. Hence:
∈ R\{ } n 1
≥
P
t B 0 , λ(t) < 1.
R
∀ ∈ \{ } | |
Third the function v () := 1 λ() is in m B ,C), and thanks to i λ(0) = m~ = m~ ~e ,
0 · − · Cb R − ∇ k k 1
we have
(cid:0) ∂v
0
j 2,...,d , (0) = 0.
∀ ∈ { } ∂x
j
Finally, up to reduce R, we deduce from (12) that there exist positive constants α, β such that:
t B , αw(t) v (t) β w(t).
R 0
∀ ∈ | | ≤ | | ≤ | |
Use of the space . Thanks to the well-known results on weak convergence of positive
• H
measures (see [4]), Theorem 1 will hold provided that we prove the following property:
+
h , lim (2πτ)d−21 ∞E f(Xn)h(Sn a(τ)) = C(L,m~,Σ,A) h(x)dx. (13)
∀ ∈ H τ + − Rd
→ ∞ n=1 Z
X (cid:2) (cid:3)
Integral decomposition. Let h be fixed and let b > 0 such that hˆ(t) = 0 when t > b.
• ∈ H k k
Nextconsideranyrealnumber ρsuchthat0 < ρ < min(R,b)andanyfunction χ (Rd,R)
∈ C∞b
compactly supported in B such that χ(t) = 1 when t ρ. For each t Rd we set
R
k k ≤ ∈
E (t) := E[f(X )ei t,Sn ]. The inverse Fourier formula gives
n n h i
(2π)dE f(X )h(S a) = hˆ(t)E (t)e i t,a dt.
n n n −h i
− Rd
Z
(cid:2) (cid:3)
6
Under Hypothesis (H), we prove below that, for every a Rd, the following series
∈
+
∞
I(a) := (2π)d E f(X )h(S a)
n n
−
n=1
X (cid:2) (cid:3)
converges, and that I(a) decomposes as the sum of three integrals called E(a), E (a) and
1
I (a). The integrals E(a) and E (a) are error terms, while I (a) is the main part of I(a). In
1 1 1
fact we have
λ(t)
I(a) = E(a)+ χ(t)hˆ(t) L(t))e i t,a dt (14)
−h i
1 λ(t)
ZBR −
with
E(a) := hˆ(t) 1 (t)χ(t) R (t)+1 (t) 1 χ(t) E (t) e i t,a dt.
ZRd (cid:18) BR n 1 n Kρ,b − n 1 n (cid:19) −h i
X≥ (cid:0) (cid:1)X≥
Next we obtain
I(a) = E(a)+E (a)+I (a) (15)
1 1
with
hˆ(t)λ(t)L(t) hˆ(0)L(0)
E1(a) := χ(t) − e−iht,aidt
1 λ(t)
Zktk≤R −
λ(t) 1 i m~ t + 1 Σt,t
+ hˆ(0)L(0) χ(t) − − k k 1 2h i e i t,a dt (16)
(1 λ(t))( i m~ t + 1 Σt,t ) −h i
Zktk≤R − − k k 1 2h i
and
χ(t)
I (a) := hˆ(0)L(0) e i t,a dt.
1 i m~ t + 1 Σt,t −h i
Zktk≤R − k k 1 2h i
Such equalities are established in [13, p. 389] (centered case). By using Hypothesis (H),
the proof of (14) borrows the same lines. To obtain (15), use the fact that for t R, t = 0,
wehave N λ(t)n 2/(b w(t)), andthefactthat1/w isintegrableat0(cf.kRke≤marks26 -3).
| n=1 |≤ | |
ProperPty (13) then follows from (15) and the next properties (17) (18) and (23).
2.2 Study of the first error term E(a)
Here we prove that we have when a + :
k k→ ∞
d 1
E(a) = o( a − −2 ). (17)
k k
For u m(Rd,C) and α = (α ,...,α ) Nd such that α := d α m , we denote by
∈ Cb 1 d ∈ | | i=1 i ≤ ⌊ ⌋
∂α the derivative operator defined by :
P
∂α ∂
∂α := | | = ∂α1...∂αd where ∂ := .
∂xα1...∂xαd 1 d j ∂x
1 d j
The following proposition is classical. Let be a bounded open subset of Rd.
O
7
Proposition 1 Let m (0,+ ) and τ = m m . Assume that u is a function in
∈ ∞ − ⌊ ⌋
⌊m⌋(Rd,C) compactly supported in and that its restriction to is in m( ,C). Then
Cb O O Cb O
the following properties hold:
(i) u m(Rd,C), and for α Nd, α = m , we have: [∂αu] = [∂αu] = [∂αu]
∈ Cb ∈ | | ⌊ ⌋ τ,Rd τ,O τ,O
(ii) C (0,+ ), a Rd, a m uˆ(a) C u + [∂αu] .
∃ ∈ ∞ ∀ ∈ k k | | ≤ k k0,Rd α= m τ,Rd
| | ⌊ ⌋
(cid:0) P (cid:1)
Proof of (17). Define:
+ +
∞ ∞
t B , (t) := R (t) and t Rd 0 , (t):= E (t),
R n n
∀ ∈ R ∀ ∈ \{ } E
n=1 n=1
X X
1 (t)hˆ(t)(1 χ(t)) (t) if t = 0
t Rd, F(t) := 1 (t)hˆ(t)χ(t) (t) and G(t) := Kρ,b − E 6
∀ ∈ BR R ( 0 if t = 0.
We have
a Rd, E(a) = Fˆ(a)+Gˆ(a).
∀ ∈
Note that χhˆ (Rd,C) is compactly supported in the closed ball B , and that, by
∈ C∞c R
Hypothesis (H), we have m(B ,C) with m > m . By applying Proposition 1 to
R ∈ Cb R d
u := F, we obtain a m Fˆ(a) = O(1), thus Fˆ(a) = o( a (d 1)/2) when a + since
− −
k k | | k k k k→ ∞
m > (d 1)/2. The same result holds for Gˆ(a) (replace B with K ). (cid:3)
R ρ,b
−
2.3 Study of the second error term E (a)
1
In this subsection we prove that we have when a + :
k k → ∞
E (a) = o( a (d 1)/2). (18)
1 − −
k k
Let E (a) andE (a) denote the two integrals in theright hand sideof (16), sothat we have:
11 12
E (a) := E (a)+E (a). Define: t B , θ (t) = χ(t) hˆ(t)λ(t)L(t) hˆ(0)L(0) . Then
1 11 12 R 1
∀ ∈ −
E (a) = q (a) with q := 1(cid:0) θ /v , (cid:1) (19)
11 1 1 BR 1 0
where v (t) = 1 λ(t). Next define: t B , θ (t) := χ(t) λ(t) 1 i m~ t + 1 Σt,t
0 − b∀ ∈ R 2 − − k k 1 2h i
and v˜ (t) := i m~ t + 1 Σt,t . Then
0 − k k 1 2h i (cid:0) (cid:1)
E (a) = q (a) with q := 1 θ /(v v˜ ). (20)
12 2 2 BR 2 0 0
Unfortunately, since q and q abre not defined at 0, q (a) and q (a) cannot be studied by
1 2 1 2
the elementary arguments of Subsection 2.2. This fact constitutes the main difficulty of the
proof in [2]. Below, we present the two key results (Probpositions 2band 3) to obtain (18). In
the next subsection, these two propositions are also used to obtain the desired result for the
main part I (a) (by difference with the Gaussian case, see Lemma 1).
1
Recallthat w() is defined in(11). Inthe two nextpropositions, we consider any real numbers
·
m > m and r > 0.
d
8
Proposition 2 Let θ and v be complex-valued functions on B such that:
r
θ m B ,C with compact support in B and θ(0)= 0,
• ∈ Cb r r
v m B ,C and
• ∈ Cb (cid:0) r (cid:1)
j 2,...,d ,(∂ v)(0) = 0 (21)
j
(cid:0) (cid:1) ∀ ∈ { }
(a,b) (R )2, x B , a w(x) v(x) b w(x). (22)
∃ ∈ ∗+ ∀ ∈ r | | ≤ | | ≤ | |
Then q := 1 θ/v is integrable on Rd and lim a (d 1)/2qˆ(a) = 0.
Br kak→+∞k k −
Proposition 3 In addition to the hypotheses of Proposition 2, we consider another complex-
valued function v˜ on B satisfying the same hypotheses as v(). Moreover we assume that all
r
·
the first and second partial derivatives of θ vanish at 0. Then q := 1 θ/(vv˜) is integrable on
Br
Rd and lim a (d 1)/2qˆ(a) = 0.
a + −
k k→ ∞k k
The proofs of Propositions 2-3, based on dyadic decompositions, are partially presented in
[1, 2]. Since dyadic decomposition is not familiar to probabilistic readers, these proofs are
detailed in Appendix A for the sake of completeness.
Proof of (18). Proposition 2 applied with θ := θ and v := v (see Rk. 3) gives q (a) =
1 0 1
o( a (d 1)/2) when a + . Similarly Proposition 3 applied with θ := θ , v := v and
− − 2 0
k k k k → ∞
v˜:= v˜0 gives q2(a) = o( a −(d−1)/2). Then (18) follows from (19) (20). b (cid:3)
k k
b
2.4 Study of the main part I (a) of I(a)
1
Let a :R+ Rd be a measurable function satisfying (9). Here we prove that
→
lim τ(d 1)/2I (a(τ)) = (2π)(d+1)/2C(L,m~,Σ,A) hˆ(0). (23)
τ + − 1
→ ∞
The proof of (23) in [2] involves the modified Bessel functions and some related computations
partially made in the book [26]. Here we present a direct and simpler proof of (23) based on
the next proposition. We denote by S(Rd) the so-called Schwartz space.
Proposition 4 Let w~ Rd 0 , and let p : R+ Rd such that P := lim p(τ)/√τ
τ +
exists in Rd. Then we h∈ave fo\r{al}l function k S(R→d) → ∞
∈
d 1 k(u)e−ihu,τw~+p(τ)i 2πd+21 P 2 w~ 2 P,w~ 2
lim τ −2 du = k(0)exp k k k k −h i .
τ→+∞ ZRd −ihw~,ui+kuk2 kw~k (cid:18)− 4kw~k2 (cid:19)
The proof of Proposition 4 (again based on Propositions 2-3) is presented below. Let us first
apply Proposition 4 to establish (23).
Proof of (23). Since m~ = m~ e~ , one can rewrite (9) as follows: a(τ) = τ m~ e~ +√τ b(τ)
1 1
k k k k
with b : [0,+ ) Rd such that lim b(τ) = A. Denote by λ ,...,λ the (positive)
τ + 1 d
∞ → → ∞
eigenvalues of Σ. Let ∆ := diag(√λ ,...,√λ ) and let P be any orthogonal d d-matrix
1 d
×
such that
P 1ΣP = ∆2 = diag(λ ,...,λ ).
− 1 d
9
Observe that Σt,t = ∆2P 1t,P 1t = ∆P 1t 2. Set ~ℓ := ∆ 1P 1e~. By using the
− − − − − 1
variable t = Ph∆ 1ui, onehobtains (use ti = kP∆ 1u,ke~ = u,~ℓ )
− 1 − 1
h i h i
I (a) = 2hˆ(0)L(0)(det∆) 1 χ(P∆−1u)e−ih∆−1u,P−1ai du.
1 − i 2 m~ ~ℓ,u + u 2
Z − h k k i k k
Set ζ(x) := χ(P∆ 1x) (x Rd) and p(τ) := √2τ∆ 1P 1b(2τ) (τ > 0). From the equality
− − −
∆ 1u,P 1e~ = u,~ℓ , it f∈ollows that
− − 1
h i h i
I (a(τ)) = 2hˆ(0)L(0)(det∆) 1 χ(P∆−1u)e−i ∆−1u,P−1 τkm~ke~1+√τb(τ) du
1 − i 2(cid:10) m~ ~ℓ,u +(cid:0) u 2 (cid:1)(cid:11)
Z − h k k i k k
= 2hˆ(0)L(0)(det∆) 1 ζ(u)e−i u,2(τ2)km~k~ℓ+p(τ2) du.
−
i 2 (cid:10)m~ ~ℓ,u + u 2 (cid:11)
Z − h k k i k k
Now ∆−1 = P−1Σ−21P gives ~ℓ = P−1Σ−21e~1, so ~ℓ = Σ−12e~1 and Σ−21m~ = m~ ~ℓ .
k k k k k k k kk k
Moreover, we have ζ(0) = χ(0) = 1 and
lim p(τ)/√τ = P with P := √2∆−1P−1A= √2P−1Σ−21 A.
τ +
→ ∞
From Proposition 4, applied with P previously defined, w~ := 2 m~ ~ℓ = 2P−1Σ−21m~, and
k k
finally with the function k := ζ, one obtains:
d+1
τ→lim+∞(τ2)d−21I1(a(τ)) = 2hˆ(0)L(0)(detΣ)−21 2 2Σπ−122m~
k k
Σ−12m~ 2 Σ−12A 2 Σ−12m~,Σ−12A 2
exp k k k k −h i ,
× (cid:18)− 2 Σ−12m~ 2 (cid:19)
k k
from which we easily deduce (23). (cid:3)
Proof of Proposition 4. Let U be an isometric linear map on Rd such that U(w~) = w~ e~ .
1
k k
Let τ > 0. The change of variable v = (v ,v ) = U(u) in the integral of Proposition 4 gives
1 ′
k(u)e i u,τw~+p(τ) k(U 1(v))e iτ w~ v1e i v,U(p(τ))
−h i − − k k −h i
du = dv,
Rd i w~,u + u 2 Rd i w~ v1+ v 2
Z − h i k k Z − k k k k
and by hypothesis we know that lim U(p(τ))/√τ = U(P). Set U(P) := (ℓ ,ℓ) with
τ + 1 ′
ℓ R et ℓ Rd 1. As U is isomet→ric,∞we have U(P) = P and ℓ = U(P),e~ =
1 ′ − 1 1
∈ ∈ k k k k h i
P,U 1(e~ ) = P,w~ / w~ . Thus
− 1
h i h i k k
ℓ 2 = P 2 P,w~ 2/ w~ 2 = w~ 2 P 2 P,w~ 2 / w~ 2.
′
k k k k −h i k k k k k k −h i k k
(cid:0) (cid:1)
Next, let us set U(p(τ)) := u(τ) = (u (τ), ,u (τ)), and u(τ) = (u (τ), ,u (τ)). Then
1 d ′ 2 d
··· ···
k(u)e−ihu,τw~+p(τ)i k(U−1(v))e−i(τkw~k+u1(τ))v1e−ihu′(τ),v′i
du = dv.
Rd i w~,u + u 2 Rd i w~ v1+ v 2
Z − h i k k Z − k k k k
10